If the whole back easily comes off, it's pretty obvious to pull it all apart and just glue the whole thing back together. Assuming you've got a chair with a bowed back that isn't loose, I'd use a glue injector. If there's not enough space around the spindles for the glue to flow in, I'd drill a couple of very small holes along the edges of the spindles (back side of chair) to let the glue flow in.

If you need to, take a picture of the chair assembled. Just kidding. Disasemble as much as possible. Scrape off the old glue to show some new wood on the ends of the spindles and in the recesses. Use Elmers or equalivent yellow woodworking glue, do the following. Put glue on the spindles and holes and assemble on a table. Use strap clamps, ropes and tourniquet system to pull it all together at one time. Doing this on a table will insure that the legs will meet the floor at the same time. Do not use fast drying glue such as "gorilla glue". It will dry over night. A weight on the chair will help, and that's up to you. Redd Foxx is out there somewhere!

artie67 has pretty much everything I would suggest in his post. Only thing I would add is that if any of the spindles or rung are loose in the sockets, wrap with some thin, glue coated cloth to build up the dowel end slightly. The woodworking glue is not good at filling gaps and creating a strong bond.

All of the spindles are "broken loose" but don't have any space around them., if that makes sense. They will have to be pounded out and it will be hard to reinsert them, I would think. I wonder if this would work well:

• remove them and cut a slot in the end of each spindle
• glue them back with Elmer's
• hammer in a wedge like in an axe handle (I think this is used in some furniture)

If it's not valuable then remove the shafts that you can most easily pull out, put some wood glue in the slots and slot them back in. Then clamp it as best you can so that the spindles are wedged tightly in place and use finishing nails (brads, as Plumking suggested) on the opposing side of each leg (or whatever piece each spindle is stuck into), aiming parallel to and centered on each spindle to secure them.

Dip the tip of each nail in mineral oil to help avoid splitting the wood. Use a finishing tool (sinker) to sink each nail below the surface of the wood. Use a dot of wood-filler to fill the holes after.

5- or 10-minute epoxy (liquid/gel, not the putty) applied to wood that's had all the old glue sanded off will be much stronger than wood glue and will fill gaps very nicely.

Use sandpaper and a small file to make sure there's raw wood exposed on both the dowel and the hole it goes into. Do all the dowels, and all the holes, at the same time or else you'll be forever repairing "the weakest link." A 15-minute, permanent fix.